Published in ACM TRANSACTIONS ON ALGORITHMS 1:1, 160-176 (2005) The NP-Completeness Column DAVID S. JOHNSON AT&T Labs { Research, Florham Park, New Jersey Abstract. This is the 24th edition of a column that covers new developments in the theory of NP-completeness. The presentation is modeled on that which M. R. Garey and I used in our book \Computers and Intractability: A Guide to the Theory of NP-Completeness," W. H. Freeman & Co., New York, 1979, hereinafter referred to as \[G&J]." Previous columns, the first 23 of which appeared in J. Algorithms, will be referred to by a combination of their sequence number and year of appearance, e.g. \[Col 1, 1981]." This edition of the column describes the history and purpose of the column and the status of the open problems from [G&J] and previous columns. Categories and Subject Descriptors: F.1.3 [Computation by Abstract Devices]: Complexity Classes|reducibility and completeness; relations among complexity classes; F.2.0 [Analysis of Algorithms and Problem Complexity]: General General Terms: Algorithms, Theory Additional Key Words and Phrases: NP-completeness, open problems, primality testing, perfect graphs, coding theory, lattice bases 1. A BELATED REVIVAL With this article, I resume a long-dormant column on NP-completeness whose first 23 editions appeared in J. Algorithms from 1981 through 1992. When the column first appeared, just two and a half years after the publication of [G&J], its main purpose was to provide timely additions and updates to the list of NP-complete and open problems at the end of that book. As the column evolved, however, it tended to devote more of its effort to providing brief reports and tutorials on new theoretical developments related to NP-completeness, covering such topics as Levin's concept of \random NP" [Col 11, 1984], the complexity of \uniqueness" [Col 15, 1985], zero-knowledge proofs [Col 21, 1988], and the PCP theorem [Col 23, 1992]. The revived column will contain material of both types, and in addition will provide pointers to other relevant sources of information as they appear, including books, tutorial articles, and websites. The revival of the column was inspired in part by the creation of the new ACM Transactions on Algorithms as a successor to J. Algorithms. In addition, I hope that the research I do in preparing the column will help me make progress on a planned 2nd edition of [G&J]. Much has happened in the 13 years since the last Author's address: Room C239, AT&T Labs - Research, 180 Park Avenue, Florham Park, NJ 07932, e-mail: [email protected]. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permission may be requested from Publication Dept., ACM, Inc., 1515 Broadway, New York, NY 10036 USA, fax: +1 (212) 869-0481, or [email protected]. c 2005 ACM 0004-5411/20YY/0100-0001 $5.00 ACM Transactions on Algorithms, Vol. V, No. N, Month 20YY, Pages 1{0??. 2 David Johnson · Table I. The current status of the open problems from [G&J] and previous columns. Problem Name Source Status Covered in GRAPH ISOMORPHISM [G&J] Open { SUBGRAPH HOMEOMORPHISM [G&J] P [Col 19, 1987] (FOR A FIXED GRAPH H) GRAPH GENUS [G&J] NPC [Col 21, 1988] CHORDAL GRAPH COMPLETION [G&J] NPC [Col 1, 1981] CHROMATIC INDEX [G&J] NPC [Col 1, 1981] PARTIAL ORDER DIMENSION [G&J] NPC [Col 1, 1981] PRECEDENCE CONSTRAINED [G&J] Open { 3-PROCESSOR SCHEDULING LINEAR PROGRAMMING [G&J] P [Col 1, 1981] TOTAL UNIMODULARITY [G&J] P [Col 1, 1981] SPANNING TREE PARITY PROBLEM [G&J] P [Col 1, 1981] COMPOSITE NUMBER [G&J] P This Column MINIMUM LENGTH TRIANGULATION [G&J] Open { IMPERFECT GRAPH [Col 1, 1981] P This Column GRAPH THICKNESS [Col 2, 1982] NPC [Col 5, 1982] EVEN COVER [Col 3, 1982] NPC This Column (MINIMUM WEIGHT CODEWORD) \UNRESTRICTED" TWO-LAYER [Col 5, 1982] Open { CHANNEL ROUTING GRACEFUL GRAPH [Col 6, 1983] Open { ANDREEV'S PROBLEM [Col 17, 1986] Open { SHORTEST VECTOR IN A LATTICE [Col 18, 1986] \NPC" This Column column appeared (and the 26 years since the first edition of [G&J]). As with the very first column [Col 1, 1981], this edition of the column surveys developments with respect to the open problem list in [G&J], this time augmenting the coverage to include the open problems highlighted in previous columns. Table I summarizes the current status of all these problems. Eight of the twelve open problems from [G&J] and one of the seven open problems from the columns had been resolved by 1992, and their resolutions were covered in previous columns. Since then one of the four open problems from [G&J] and two of the open problems from the columns have been resolved, and one of the column problems has been partially resolved (in a sense to be explained later). Section 2 will cover the resolved and partially resolved problems, while Section 3 will discuss the problems that remain open. The next column will likely cover hardness-of-approximation results and the complexity conjectures on which they rely. Suggestions of topics and results to be covered by future columns are welcome. While readers await the next column, they might wish to investigate some of the many other sources that now provide information about developments in the field. ACM Transactions on Algorithms, Vol. V, No. N, Month 20YY. The NP-Completeness Column 3 · SIGACT News has been running a Computational Complexity column moderated by Lane Hemaspaandra since 1993, available in .pdf format from the ACM Digi- tal Library (portal.acm.org/dl.cfm). Although the column covers a wide range of topics, many are directly relevant to NP-completeness. In particular, the 46th edition, which appears in the March 2005 issue and was written by guest columnist Scott Aaronson, is a delightful survey of the wide variety of proposals for using physical processes to obtain exponential speedups over classical Turing machines and thus solve NP-complete problems efficiently. It covers a wide variety of sug- gestions, from soap bubbles to various exploitations of quantum mechanics, and explains why each is unlikely to work. A second relevant column has been appearing regularly in the Bulletin of the EATCS since 1987. Originally entitled \The Structural Complexity Column" and moderated by Juris Hartmanis, it morphed in 1997 into \The Complexity Column," moderated first by Eric Allender, then by Lance Fortnow, and currently by Jacobo Tor´an. An index and downloadable .pdf versions of recent editions are available from http://theorie.informatik.uni-ulm.de/Personen/toran/beatcs. The October 2004 edition presents an interesting survey by J¨org Flum and Martin Grohe on the relatively new concept of “fixed parameter tractability" and its associated complexity classes (see also Downey and Fellows [1999]). There are also several regularly updated websites/blogs that may be of interest. Lance Fortnow has been writing a \Computational Complexity" weblog since Au- gust 2002, with daily updates. This is where many of us first hear about major results, and where we could even find technical reviews of the early episodes of the CBS television series Numb3rs, in the second episode of which the question of P ver- sus NP played a central role. A webpage maintained by Scott Aaronson, \The Com- plexity Zoo" (http://www.complexityzoo.com) provides notation and definitions for hundreds of complexity classes, both common and obscure, along with some of the facts known about them. Pierluigi Crescenzi and Viggo Kann maintain an \NP Optimization Problem" website (http://ww.nada.kth.se/ viggo/problemlist /compendium.html), which collects hardness-of-approximation∼ results, updated at least through March 2000. Gerhard Woeginger maintains \The P-versus-NP" page (http://www.win.tue.nl/ gwoegi/P-versus-NP.htm) with many interest- ing links plus a list of supposed pro∼ ofs that P = NP (and P = NP), all but one of which appeared since the brief survey of such claims in [Col 20,6 1987]. Indeed, 17 of the 19 claims in the list have occurred since the year 2000, when the Clay Institute announced prizes of $1,000,000 for resolving the P versus NP question and six other famous problems in mathematics (see http://www.claymath.org/millennium). Finally, readers who have not seen the earlier editions of this column (or have forgotten them) can now obtain them online. I recently compiled .pdf versions of all 23 columns from the original troff source files, and have posted them at http//www.research.att.com/ dsj/columns. These are inexact replicas, with slightly different pagination and∼some subpar equation formatting due to changes in the underlying typesetting software. In addition, the figures in Columns 5 and 16 had to be recreated because the software that originally generated them was no longer functional. (Oh, the joys of software evolution!) Definitive electronic versions of the columns can be found at Science Direct (http://www.sciencedirect.com), ACM Transactions on Algorithms, Vol. V, No. N, Month 20YY. 4 David Johnson · where the J.